A014664 -id:A014664 - cp's OEIS frontend

A211245 Order of 9 mod n-th prime: least k such that prime(n) divides 9^k-1.

1, 0, 2, 3, 5, 3, 8, 9, 11, 14, 15, 9, 4, 21, 23, 26, 29, 5, 11, 35, 6, 39, 41, 44, 24, 50, 17, 53, 27, 56, 63, 65, 68, 69, 74, 25, 39, 81, 83, 86, 89, 45, 95, 8, 98, 99, 105, 111, 113, 57, 116, 119, 60, 125, 128, 131, 134, 15, 69, 140, 141, 146, 17, 155, 39

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A364867.

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A002371, A372801.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(9,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 9; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=9}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

From Jianing Song, May 13 2024: (Start)
a(n) = A062117(n)/gcd(2, A062117(n)).
a(n) <= (prime(n) - 1)/2. Those prime(n) for which a(n) = (prime(n) - 1)/2 are listed in A364867. (End)

A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

Original entry on oeis.org

1, 1, 4, 0, 10, 12, 16, 3, 22, 7, 15, 9, 40, 6, 23, 26, 29, 60, 66, 70, 24, 78, 41, 88, 96, 100, 51, 106, 27, 14, 126, 65, 68, 69, 74, 150, 52, 162, 83, 172, 178, 12, 10, 24, 98, 99, 210, 37, 113, 228, 116, 238, 240, 125, 256, 262, 268, 135, 138, 20, 141, 292

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A019337 (full reptend primes in base 7).
In other bases: A014664, A062117, A082654, A211241, A211242, A211244, A211245, A002371.
Row lengths of A201911. - Michel Marcus, Feb 04 2019

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(7,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 7; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=7}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

Original entry on oeis.org

0, 2, 4, 1, 10, 4, 8, 6, 11, 28, 5, 12, 20, 14, 23, 52, 58, 20, 22, 35, 3, 13, 82, 11, 16, 100, 17, 106, 12, 28, 7, 130, 68, 46, 148, 5, 52, 54, 83, 172, 178, 60, 95, 32, 196, 33, 70, 37, 226, 76, 29, 119, 8, 50, 16, 131, 268, 45, 92, 70, 94, 292, 34, 155, 52

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053451 (order of 8 mod 2n+1), A019338 (full reptend primes in base 8).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211245, A002371, A372801.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(8,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 8; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=8}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

a(n) = A014664(n)/gcd(3, A014664(n)). - Jianing Song, May 13 2024

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20, 46, 22, 47, 21, 15, 9, 57, 42, 15, 48, 28, 41, 48, 60, 85, 25, 74, 25, 52, 11, 32, 51, 17, 13, 34, 113, 13, 71, 2, 16, 64, 130, 81, 35, 37, 29, 39, 147, 68, 60, 71, 96, 92, 99, 12

Offset: 2

Felix Fröhlich, May 28 2015

nonn

p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.

Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?

Felix Fröhlich, Table of n, a(n) for n = 2..10000
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.

Cf. A001220, A195988, A241014, A244801, A246568, A250406, A250407, A306885, A338646, A343410.

a(n,p=prime(n))=abs(centerlift(Mod(2,p^2)^((p-1)/2))\/p)
apply(p->a(0,p), primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 15 2015

a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - Daniel Chen, Sep 01 2022

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A139686 Odd multiplicative orders of 2 modulo primes.

Original entry on oeis.org

3, 11, 5, 23, 35, 9, 39, 11, 51, 7, 15, 83, 95, 99, 37, 29, 119, 131, 135, 155, 21, 179, 183, 191, 43, 73, 231, 239, 243, 251, 299, 25, 303, 45, 323, 359, 121, 371, 375, 411, 419, 431, 55, 443, 91, 153, 117, 483, 491, 495, 515, 519, 531, 543, 29, 575, 611, 615, 639

Offset: 1

Max Alekseyev, Apr 29 2008

nonn

Subsequence of A014664, consisting of odd elements.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014664, A014663 (corresponding primes).

Cf. other bases: this sequence (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

p = Select[Range[1000], PrimeQ]; Select[MultiplicativeOrder[2, #] & /@ p, OddQ] (* Amiram Eldar, Jul 30 2020 *)

forprime(p=3,10^5,z=znorder(Mod(2,p));if(z%2,print1(z,", ")))

a(n) = multiplicative order of 2 modulo A014663(n).

A337878 a(n) is the smallest m > 0 such that the n-th prime divides Jacobsthal(m).

Original entry on oeis.org

3, 4, 6, 5, 12, 8, 9, 22, 28, 10, 36, 20, 7, 46, 52, 29, 60, 33, 70, 18, 78, 41, 22, 48, 100, 102, 53, 36, 28, 14, 65, 68, 69, 148, 30, 52, 81, 166, 172, 89, 180, 190, 96, 196, 198, 105, 74, 113, 76, 58, 238, 24, 25, 16, 262, 268, 270, 92, 35, 47, 292, 51

Offset: 2

A.H.M. Smeets, Sep 27 2020

nonn

All positive Jacobsthal numbers are odd, so the index starts at n = 2.
The set of primitive prime factors of J_k is given by {A000040(j) | a(j) = k}.
By definition, a(n) is the multiplicative order of -2 modulo the n-th prime for n > 2. - Jianing Song, Jun 20 2025

			The 4th prime number is 7, and 7 divides 21 which is Jacobsthal(6), so a(4) = 6. The second prime number, 3, divides Jacobsthal(6) as well, but it divides also the smaller Jacobsthal(3), i.e., a(2) = 3.

Jianing Song, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Primitive Prime Factor

Cf. A000040 (primes), A001045 (Jacobsthal numbers), A001602 (similar for Fibonacci numbers), A105874 (primes having primitive root -2), A129738.
Cf. multiplicative orders of 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. multiplicative orders of -2..-10: this sequence (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

m = 300; j = LinearRecurrence[{1, 2}, {3, 5}, m]; s = {}; p = 3; While[(ind = Select[Range[m], Divisible[j[[#]], p] &, 1]) != {}, AppendTo[s, ind[[1]] + 2]; p = NextPrime[p]]; s (* Amiram Eldar, Sep 28 2020 *)

J(n) = (2^n - (-1)^n)/3; \\ A001045
a(n) = {my(k=1, p=prime(n)); while (J(k) % p, k++); k;} \\ Michel Marcus, Sep 29 2020

n = 1
while n < 63:
    n, J0, J1, a = n+1, 3, 1, 3
    p = A000040(n)
    J0 = J0%p
    while J0 != 0:
        J0, J1, a = (J0+2*J1)%p, J0, a+1
    print(n,a)

A000040(n) == 1 (mod a(n)) for n > 2.

A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

Original entry on oeis.org

1, 0, 4, 3, 10, 6, 16, 9, 22, 28, 15, 9, 8, 21, 46, 52, 58, 5, 11, 70, 12, 39, 82, 88, 48, 100, 17, 106, 54, 112, 63, 130, 136, 69, 148, 25, 39, 81, 166, 172, 178, 90, 190, 16, 196, 99, 105, 111, 226, 114, 232, 238, 120, 250, 256, 262, 268, 15, 138, 280

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105875 (primes having primitive root -3).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), this sequence, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

A380482[n_] := If[n == 2, 0, MultiplicativeOrder[-3, Prime[n]]];
Array[A380482, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-3}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

Original entry on oeis.org

0, 2, 1, 6, 10, 3, 4, 18, 22, 7, 10, 9, 5, 14, 46, 13, 58, 15, 66, 70, 18, 78, 82, 22, 24, 25, 102, 106, 9, 7, 14, 130, 17, 138, 37, 30, 13, 162, 166, 43, 178, 45, 190, 48, 49, 198, 210, 74, 226, 19, 58, 238, 12, 50, 8, 262, 67, 270, 23, 70

Offset: 1

Jianing Song, Jun 27 2025

a(n) divides (p-1)/4 if p = prime(n) == 1 (mod 4), since (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i^2 == -1 (mod p).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105876 (primes having primitive root -4).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, this sequence, A380532, A380533, A380540, A380541, A380542, A385222.

A380531[n_] := If[n == 1, 0, MultiplicativeOrder[-4, Prime[n]]];
Array[A380531, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-4}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380532 a(n) is the multiplicative order of -5 modulo prime(n); a(3) = 0 for completion.

Original entry on oeis.org

1, 1, 0, 3, 10, 4, 16, 18, 11, 7, 6, 36, 20, 21, 23, 52, 58, 15, 11, 10, 72, 78, 41, 44, 96, 50, 51, 53, 54, 112, 21, 130, 136, 138, 74, 150, 156, 27, 83, 172, 178, 30, 38, 192, 196, 66, 70, 111, 113, 57, 232, 238, 40, 50, 256, 131, 134, 54, 276, 140

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105877 (primes having primitive root -5).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, this sequence, A380533, A380540, A380541, A380542, A385222.

A380532[n_] := If[n == 3, 0, MultiplicativeOrder[-5, Prime[n]]];
Array[A380532, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-5}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

cp's OEIS Frontend

A211245 Order of 9 mod n-th prime: least k such that prime(n) divides 9^k-1.

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A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

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A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

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A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

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A086251 Number of primitive prime factors of 2^n - 1.

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A139686 Odd multiplicative orders of 2 modulo primes.

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A337878 a(n) is the smallest m > 0 such that the n-th prime divides Jacobsthal(m).

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